Labeling Isometric and Almost Isometric n-Point Configurations in R

Consider two n-point configurations X and Y in R with the same distance distribution and distinct distances. We begin with proposing an algorithm that tries to find a bijection ψ : X → Y such that X and Y are congruent under ψ, or determines that no such bijection exists. Next, we discuss extending this algorithm to the case where X and Y have small multiplicities of recurring distances, then to the case where X and Y have “almost the same” distance distribution.